Method and system for target localization

ABSTRACT

The present inventions comprise aA method of estimating a minimum range for a target with respect to a first point of interest, independent of actual, range to the target, comprising obtaining three bearing data points; using the three bearing data points to determine a speed contribution V os  cos (θ β ) of a first point of interest to a distance from a relative velocity vector over a time frame comprising t 0  to t 0 ′; determining an angle θ β  as defined by the bearing relative to ownship&#39;s heading at the point in time of closest approach to a second point of interest; and calculating a minimum range using a predetermined formula.

FIELD OF THE INVENTION

The present inventions relate to localization of an object or target ofinterest.

DESCRIPTION OF THE RELATED ART

It is often desirable to track one object from another object todetermine if the tracked object will intercept the tracking object, orat what point in time will the tracked object be at it closest approachto the tracking object, sometimes referred to in the art as “TargetMotion Analysis.” For example, a vessel afloat in the presence of subseaor partially submerged obstacles would need to know where thoseobstacles are in order to avoid hitting those obstacles. By way ofexample and not limitation, such systems have been proposed in the artto avoid collisions with other vessels, collisions with such asicebergs, and collisions with submerged objects sufficient to causedamage such as ledges, seamounts, or reefs.

Some of the prior art has proposed using statistically based trackingmethods. For example, U.S. Pat. No. 5,732,043 to Nguyen et al. for“Optimized Deterministic Bearings Only Target Motion Analysis Technique”teaches using four target bearings to optimize a target track solution.

In other art, U.S. Pat. No. 6,199,471 issued to Perruzzi, et al. for a“Method And System For Determining The Probable Location Of A Contact”teaches a method and a system for determining a weapon firing strategyfor an evading target. Perruzzi '471 comprises the steps of sensing themotion of the target, analyzing the motion of the target, providing aweapon employment decision aid, determining the evasion region for thetarget using the weapon employment decision aid and the analyzed motion,visually displaying the evasion region, feeding operator knowledge aboutevading target, and generating a representation of the probability ofthe location of the evading target.

U.S. Pat. No. 5,867,256 to Van Rheeden for “Passive Range EstimationUsing Image Size Measurements” teaches a range estimation system andmethod which comprises a data base containing data for identification ofcertain targets and data for estimating the initial range to each of thetargets as a function of the observed dimensions of the targets. Asensor (1) observes a scene containing a target a plurality of spacedapart times while the sensor is moving relative to the target to providedata from each observation of the scene relating to the dimensions ofthe target within the scene. The remaining range to the target isestimated from the observed dimensions of the target from the rangetraveled since a prior estimation of range and from a prior estimationof the remaining range to the target. The sensor (1) provides electricalsignals representing the observed scene (3) and can be a visible lightor infrared sensor. A computer (9) is used to identify the target fromthe data base, estimate the initial range to the target and estimate theremaining range from the range traveled between successive observationsof the scene and the change of dimensions of the target in the observedscene.

As noted in the prior art, there are a number of situations where it isdesirable to estimate the range to an object of interest or target (e.g.aircraft without the aid of instrument landing systems, automobiles thatwould be aware of the distance between vehicles to avoid collisions, andmissile-based warfare). As also known in the art, active techniques tomeasure range, such as radar, ladar and sonar, have drawbacks, primarilyin military applications, including easy detection by the target underattack. This is true, for example, in submarine warfare where one vesselmay want to use sonar to determine the position and velocity of an enemyship. In such situations, it is advantageous to estimate range to thetarget passively.

For passive tracking situations, in order to react quickly, trackingmethods would preferably fix a boundary on the range to the trackedobject quickly while using a minimum amount of data, preferably passivedata. Further, it is preferable to calculate the bearing of the trackedobject with respect to the tracking object at a point of closestapproach, along with calculating a time to that closest approach,independent of other position data.

The AN/SQQ-89(V) UFCS (Navy) surface ship ASW Fire Control Systemcurrently uses the Manual Adaptive Target Estimator (MATE) and MaximumLikelihood Estimator (MLE) algorithms to determine target position.These algorithms require substantially more data than the presentinventions to obtain their results. The MATE algorithm requires operatorbased estimates, and systematic manual manipulation of the data toarrive at a position, course and speed estimate of the target. The MLEalgorithm also requires limited operator input to arrive at astatistically based estimate of position, course and speed of thetarget. Both of these algorithms require a substantial amount of data,approximately fifteen to twenty data points, to arrive at a stablesolution.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the presentinventions will become more fully apparent from the followingdescription, appended claims, and accompanying drawings in which:

FIG. 1 is an exemplary Cartesian plot of a target, an ownship, andvarious vectors related to the two, in a geographic reference frame; and

FIG. 2 is an exemplary Cartesian plot of a target, an ownship, andvarious vectors related to the two, in a reference frame relative to anownship's position;

FIG. 3 is an exemplary Cartesian plot showing determination of targetmaneuvers and noise in the system; and

FIG. 4 is a schematic representation of an exemplary system.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, an exemplary Cartesian plot of a target, anownship, and various vectors related to the two in a geographicreference frame, the present inventions comprise a method of providingbounds for approximations for tracking an object such as target 2 withrespect to a first object such as ownship 1. The present inventionscomprise methods for creating calculations useful for bounding trackingsensor localization using a substantially minimum amount of data, in apreferred embodiment especially using passively obtained data as thatterm is understood by those of ordinary skill in the target detectionarts. The methods comprise calculating relative bearing at a closetpoint of approach (“CPA”) and time of CPA independently of otherposition data, estimating target motion analysis (“TMA”) solution noise,and detecting contact maneuvers.

In a preferred embodiment, the methods of the present inventions may beused to conduct passive TMA using symmetries associated with twodifferent views of a problem to be solved, e.g. two reference frames andtwo points of interest. A first of these frames, geographic frame ofreference 100, is shown in FIG. 1 and second frame of reference,relative frame of reference 200, is shown in FIG. 2.

As used herein, the “points of interest” include a first physical objectsuch as ownship 1, and a second, target 2, such as second vessel. Asfurther used herein, “ownship” means a first reference point that is nota target, i.e. the vessel making the calculations. Each of these pointsof interest may be in motion or stationary, and, if in motion, may be inmotion in different planes with respect to each other. “Target motionanalysis” or TMA means that the course and speed for target 2, which mayinitially be unknown, are resolved as well as the range to and bearingof target 2 at or for a predetermined time frame with respect to ownship1. In a preferred embodiment of the present inventions, bearing at CPA,time of CPA, a minimum range to the target with associated course andspeed for the minimum range only as a limiting condition, and an initialestimate of the target's true range, course and speed may be determined.

The methods of the present inventions are not limited to surface orsubsea water vessels. By way of example and not limitation, target 2 maybe another vessel, an iceberg, a submerged object such as a ledge orreef, or the like, assuming that target 2 emits a signal that can bedetected by a passive sensor for the passive solution. Further, themethods of the present inventions may be used with partially or fullysubmerged features such as rocks or debris, floating materials,stationary materials, and the like, or combinations thereof, especiallyif the presence of such features may be determined, but a measurement ofrange to the feature may be lacking in the detection device that detectsthe feature. However, it is expressly understood that active as well aspassive data may be used in the present inventions' methods, in whichcase any single active signal may be used to determine a range valuewhich can then be used in conjunction with passive data to fully resolverange, bearing, course and speed.

In general, the present inventions' methods comprise obtaining at leastthree bearing and time data points for a first estimate, e.g. at timepoints t₁, t₂, t₃, t₄. These data are used to isolate a passive TMAestimate based on a single leg of time tagged, bearings only data, i.e.no maneuvering of the first point of interest such as ownship 1 isrequired to obtain a passive estimate. Further, the present inventions'methods comprise a closed form expression for an estimate that may beresolved in a single iteration as opposed to prior art methods such asthose using first order statistical solutions.

The present inventions' methods utilize velocity vectors of the twoitems of interest, i.e. vector 13 and estimated vector 30. Thesevelocity vectors, when arranged to determine their vector difference,form one side 52, 53 of a parallelogram as well as a diagonal of thatparallelogram, shown as darkened portion 51 of vector 13. For theparallelogram to remain a parallelogram when angles of vertices of theparallelogram change, the perpendicular distances to respective oppositesides of the parallelogram change in a predetermined fashion, i.e. asthe angles of the parallelogram whose diagonal remains at substantiallythe same orientation to ownship 1's constant course, change from π/2,the corresponding length of the diagonal must increase by an amountequal to the relative velocity of ownship 1 and target 2 multiplied bythe new elapsed time value for the second course crossing minus t₀, suchthat perpendicular distance to opposing sides increases by an amountproportional to twice the range at CPA. Additionally, the greater thedifference between values of adjacent vertices, the smaller theperpendicular distance to opposing sides.

Further, successive time-lagged bearing lines, e.g. lines 11 and 12,form a parabola, shown as solution parabola 15, in geometric referenceframe 100 for substantially all geometries involving two points ofinterest 1,2, where each of the points of interest 1,2 maintains asubstantially constant respective course and speed over a time periodused for obtaining bearing measurements. Solution parabola 15 is formedby recognizing that each of the bearing lines 11,12,13,20,30 ingeographic reference frame 100 are tangent to solution parabola 15 at apredetermined, unique point. If the bearing lines of a data setbelonging to one target are tangent to solution parabola 15 at variouspoints along solution parabola 15, and if the angles of theparallelogram vertices change such that the angle of course incidencedeviates from the value at which the relative velocity vector bisectsthe angle of course incidence and the courses represented by two of theparallelogram sides are constrained to remain tangent to theparallelogram, the perpendicular distance to opposing sides alwaysincreases. This increase may only be accomplished by increasing theparallelogram perimeter.

Accordingly, solution parabola 15 will be fixed in geographic referenceframe 100, and each data set to be gathered will generate one and oneonly solution parabola 15, although different data sets may generate thesame solution parabola 15. Further, for all potential pairs of bearinglines 11,12,13,20,30 tangent to solution parabola 15 when the course ofownship 1 is one of the bearing lines and remains fixed, e.g. line 13,the value of the bearing at the CPA, e.g. angle 50′, is constant forpotential ranges at CPA. As a result, the difference vector of eachpotential velocity vector pair, i.e. velocity vector for target 2 andvelocity vector of ownship 1, remains parallel for all geometriesinvolving those two points of interest where each point of interest 1, 2maintains its respective course and speed at a constant value during thetime of measurements and calculation. This allows calculation of bearingat CPA, time of CPA, and minimum range at CPA, with data comprising asingle leg of passive, time tagged bearings. Further, this allowsestimates of TMA solutions based on minimum range and preferred rangeestimates with data comprising a single leg of passive, time taggedbearings.

Referring now to FIG. 2, to help ensure that solution parabola 15 isfixed at the correct location in geographic reference frame 100, thepresently preferred embodiment of the present inventions' methodsrequires fixing an ownship 1 at rest reference frame 200 with respect togeographic reference frame 100. In the preferred embodiment, this may beaccomplished by requiring that the location of ownship 1 at an initialtime t₀ is the same point in the two reference frames, e.g. 10, and thatthe bearing value BRG₀ is equal to zero (as used herein “BRG” meansbearing).

In the case where the incident angle of the mutual courses of target 2and ownship 1 is greater than π/2, an additional step may be required toreflect the original bearing line data, e.g. 13, around a preferredbearing line in the original data set indicated by the axis of originalsolution parabola 15 to generate revised parabola 15 for a set ofpseudo-data that reflects the course of target 2 in a reference framefor which the incident angles of courses is less than π/2. Thissituation will also require extrapolating the course of ownship 1 into apredetermined future time point and reversing the course such that theownship arrives at the same point at the time ownship 1 crosses thecourse of target 2.

Referring additionally to FIG. 1, ownship 1 is located initially atpoint 10. In the preferred embodiment, a first step to calculation ofsolution parabola 15 is to obtain three bearing data points, e.g. attimes t₁,t₂,t₃,or t₄, wherein the times t₁,t₂,t₃, or t₄ at which thebearing data points were obtained are also obtained. Bearing data iscollected in a fixed ownship reference frame such as frame 100. At aminimum, three bearing-time data points are obtained that are relativebearings with respect to point 10.

Bearing data may then be translated to a moving ownship reference frame200. Two sets of data may form vectors, one set representing target 2,e.g. 30, and the other set representing ownship 1, e.g. 13, which maythen cross each other at different times. By way of example and notlimitation, vectors 30 and 13 may cross when target 2 appears at 0°relative bearing or 180° known bearing, or when ownship 1 appears at 0°relative to the course of target 2 or when ownship 1 appears at 180°unknown to the course of target 2.

As will be understood, a large, potentially infinite number of potentialsolution points may exist based on passive bearing data. Accordingly,the present inventions' method selects at least one potential solutionpoint, e.g. bearing line 20, to indicate a range at CPA. In a preferredembodiment, bearing line 20 may be selected manually by examining targetgeometry. In alternative embodiments, bearing line 20 may be selectedautomatically such as by using artificial intelligence methods,heuristics, or the like, or a combination thereof.

Referring back to FIG. 1, once the initial three bearing data areobtained, a first estimate may be computed for relative bearing at CPA,as well as a time of CPA, by the following formulae:tan(θ_(β)−θ_(i))/=V_(REL)(t_(β)−t_(i))/R_(CPA)|θ_(i)=0  (1)t_(β)=R_(CPA)[tan(θ_(β)−θ_(i))/V_(REL)]+t_(i)|θ_(i)=0  (2)

$\begin{matrix}{\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}\Delta\; t_{j,k}} + {{\tan\left( \theta_{j} \right)}\Delta\; t_{k,i}} + {{\tan\left( \theta_{k} \right)}\Delta\; t_{i,j}}}{\begin{matrix}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}\Delta\; t_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}\Delta\; t_{k,i}} +} \\{{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}\Delta\; t_{i,j}}\end{matrix}} \right\rbrack}} & (3)\end{matrix}$In these equations (1), (2), and (3),

-   -   θ_(β) is as defined in equation (3) and representatively shown        as angle 50 in FIG. 1;    -   θ_(i) is the bearing angle to the target 2 relative to ownship 1        at time t_(i) and representatively shown as angle 50′ in FIG. 1;    -   t_(β) is the time at which θ_(β) was measured;    -   t_(i) is the time at which θ_(i) was measured;    -   Δt is the difference between two time measurements, e.g.        Δt_(j,k) is the difference between time t_(j) and time t_(k);    -   V_(REL) is the difference velocity between target 2 and ownship        1; and    -   R_(CPA) is the range to target 2 at CPA.

The formulae of the present inventions' methods may then be used tocalculate a bearing fan to determine bearing data at a predeterminedtime in the future, independent of other position data. A bearing fan isa group of bearing data spaced at predetermined points in time thatpredicts where in bearing space target 2 will be at some point in futuretime, assuming that target 2 and ownship 1 maintain their current courseand speed. By way of example and not limitation, the present inventionsmay be used to generate both relative and true bearings and time at CPA,where the time at relative bearing equals zero degrees (0°) or onehundred eighty degrees (180°).

The formulae also provide an early estimate of minimum target ranges forany bearing, independent of other position data. Further, the formulaemay be useful in many other ways, by way of example and not limitationfor providing parameters useful for early target maneuver detectors orOpen/Close determinations as well as estimates of a ratio of relativespeed to range at CPA.

The present inventions' methods may further be used to provide areal-time measure of the effect of noise on potential solutions. In apreferred embodiment, this real-time measure begins with a fourth datapoint, e.g. data point t₄.

Having selected a potential solution point, e.g. bearing line 20, thedirection of the relative velocity vector 60 can be determined.

Referring now to FIG. 4, in a preferred embodiment, data obtained forthe calculations defined herein are preferably manipulated by computer200 which has been programmed to carry out the functions set forth inthis description and typically accessible to ownship 1 such as by beingonboard ownship 1. Computer 200 may comprise any suitable computer knownin the art. Computer 200 further comprises a processor, memory, andoutput device (not shown in the figures) as well as range calculationsoftware executing within computer 200. Output device 210 may comprise adisplay device 210, a hard copy device 212, or the like, or acombination thereof.

Data sets comprising passive bearing data may be gathered such as byusing one or more sensors (shown as 230 in FIG. 4 for illustration)deployed within or near ownship 1 and capable of passively obtaining abearing to target 2 from a desired location such as ownship 1 andproviding measurements related to target 2 and ownship 1. Sensors 230may comprise any suitable sensors known in the art such as passiveacoustic sensors. The data may be passively obtained by numerous meansas will be familiar to those of ordinary skill in the passive dataacquisition arts. Once gathered, these data may be stored for laterprocessing in the memory of computer 200 or in a passive bearing datacollection device (not shown in the figures) that is addressably incommunication with the computer. The analysis performed may occur withinthe computer or a portion of the computer which has been programmed toanalyze the data received by the sensors.

Using the range calculation software, the computer may retrieve at leastthree of the stored bearing data points obtained from the bearingdetector, such as from the computer's memory. The range calculationsoftware may then use the three retrieved bearing data points todetermine a speed contribution V_(os) cos(θ_(β)) of a first point ofinterest to a distance from a relative velocity vector over a time fromt₀ to t₀′ in accordance with the teachings of the present inventions. Byway of example and not limitation, in accordance with the teachings ofthe present inventions the range calculation software may determine anangle θ_(β) defined by the bearing of target 2 relative to a heading ofownship 1 at the point in time of closest approach to a second point ofinterest and then calculates a minimum range from the source to thetarget asMin R_(CPA)=V_(os)(t_(β)−t_(i))cos(θ_(β)−θ_(i))_(θ) _(i) _(|=0); and

The range calculation software may then generate a representation of theprobability of the location of target 1 and present that informationsuch as on the output device.

In the operation of an exemplary embodiment, referring to FIG. 1 andFIG. 2, it is first noted that the following expression holds for linearmotion when an object moving in a straight line with a velocity ofV_(R), e.g. target 2, passes a stationary observer, e.g. ownship 1, at adistance of R_(CPA) where R_(CPA) is the distance at closest approach tothe stationary observer:tan(θ_(i)−θ₀)=(t_(i)−t₀)(V_(R)/R_(CPA))  (4)As used in equation (4),

-   -   θ₀ is the angle between ownship 1's heading and target 2 at an        initial time t₀;    -   θ_(i) is the angle between ownship 1's heading and target 2 at        time t_(i);    -   t_(i) is the time of bearing reading θ_(i); and    -   t₀ is the time of bearing reading θ₀.        Further, the ratio V_(R)/R_(CPA) is a calculated value, and        therefore V_(R) may be estimated based on an estimated value of        R_(CPA). Alternatively, R_(CPA) may be estimated based on an        estimated value of V_(R).

Additionally, it is noted that relative velocity vector 60 isperpendicular to the relative bearing line 20 at CPA in fixed ownshipreference frame 100, allowing for calculation of a minimum rangeestimate at CPA R_(CPA) that is substantially independent of actualcontact range. By way of example and not limitation, although at thispoint the “correct” solution may be unknown, a minimum range estimatecalculation is possible because a point when CPA occurs is known as isthe point at which target 2 is detected at relative bearing equalsθ_(β). The minimum range estimate for the distance at which ownship 1 isclosest to target 2, R_(CPA), shown in FIG. 1 at 51, may be calculatedby:Min R_(CPA)=V_(os)(t_(β)−t₀)cos(θ_(β)−θ₀)  (5)In equation (5),

-   -   t_(β) is the time at which θ_(β) was measured;    -   t₀ is the time of bearing reading θ₀;    -   V_(os) is magnitude of the velocity of ownship; and    -   θ₀ is the angle between ownship 1′s heading and target 2 at a        time t_(i)=0.

If an actual solution is selected, a right triangle may be formed byusing ownship vector 51 multiplied by the Δt_(CPA) as the hypotenuse 32of that triangle. Accordingly, the contact's range at CPA may bedetermined using hypotenuse 32, the relative bearing at CPA, and therelative velocity vector as follows:R_(CPA) _(est) =V_(OS)*Δt_(CC)*cos(θ_(β))  (6)where

-   -   Δt_(CC) is the difference between course crossings, course        crossings being defined as the time when ownship 1 crosses the        target 2's course and to and the other components have the        definitions given above.

Accordingly, using these estimates, the following calculations can thenbe made. For bearing BRG at CPA, independent of actual contact range,

$\begin{matrix}{\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}\Delta\; t_{j,k}} + {{\tan\left( \theta_{j} \right)}\Delta\; t_{k,i}} + {{\tan\left( \theta_{k} \right)}\Delta\; t_{i,j}}}{\begin{matrix}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}\Delta\; t_{j,k}} + {{\tan\left( \theta_{k} \right)}{\tan\left( \theta_{i} \right)}\Delta\; t_{k,i}} +} \\{{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}\Delta\; t_{i,j}}\end{matrix}} \right\rbrack}} & (7)\end{matrix}$In equation (7),

-   -   θ_(i) is the angle between ownship 1's heading and target 2 at        time t_(i);    -   θ_(j) is the angle between ownship 1's heading and target 2 at        time t_(j);    -   θ_(k) is the angle between ownship 1′s heading and target 2 at        time t_(k); and    -   Δt_(α,β) is the time difference between measurements θ_(α),        θ_(β) respectively, i.e., where α and β are generic indices        which are respectively pair-wise, i.e. (j,k), (k,i), and (i,j).

For the ratio of relative speed to the range at CPA,

$\begin{matrix}{\frac{V_{REL}}{R_{CPA}} = \frac{\left\lbrack {\frac{{\tan\left( \theta_{\beta} \right)} - {\tan\left( \theta_{i} \right)}}{1 + {{\tan\left( \theta_{\beta} \right)}{\tan\left( \theta_{i} \right)}}} - \frac{{\tan\left( \theta_{\beta} \right)} - {\tan\left( \theta_{j} \right)}}{1 + {{\tan\left( \theta_{\beta} \right)}{\tan\left( \theta_{j} \right)}}}} \right\rbrack}{\Delta\; t_{ij}}} & (8)\end{matrix}$In equation (8),

-   -   θ_(β) is the BRG at CPA;    -   θ_(i) is the angle between ownship 1's heading and target 2 at        time t_(i);    -   θ_(j) is the angle between ownship 1's heading and target 2 at        time t_(j); and    -   Δt_(i,j) is the time difference between measurements θ_(i) and        θ_(j).

For the time of CPA independent of actual contact range,

$\begin{matrix}{t_{\beta} = \left. {{\frac{R_{CPA}}{V_{REL}}\left\lbrack {\tan\left( {\theta_{\beta} - \theta_{i}} \right)} \right\rbrack} + t_{i}} \right|_{\theta_{i} = 0}} & (9)\end{matrix}$In equation (9),

-   -   θ_(β) is the angle between ownship 1's heading and target 2 at        CPA;    -   θ_(i) is the angle between ownship 1's heading and target 2 at        time t_(i);    -   t_(i) is the time of bearing reading θ_(i); and    -   t_(β) is the time of bearing reading θ_(β), time at which CPA        occurs.

For an estimate of the minimum range at CPA, independent of actualcontact range,MinR_(CPA)=V_(os)(t_(β)−t_(i))cos(θ_(β)−θ_(i))_(θ) _(i) _(|=0)  (10)In equation (10),

-   -   θ_(β) is the angle between ownship 1's heading and target 2 at        CPA;    -   θ_(i) is the angle between ownship 1's heading and target 2 at        time t_(i);    -   V_(os) is a magnitude of ownship's velocity;    -   t_(i) is time of bearing reading θ_(i); and    -   t_(β) is the is the time at which θ_(β) was measured.

Using these formulae, an estimate of minimum range at a predeterminedtime may therefore calculated by:Min R_(est)=Min. R_(CPA)/cos(θ_(β)−θ_(j))|θ_(j)=current bearingmeasure  (11)where the terms in equation (11) are defined above.

Further, from an estimate of R_(CPA(Minimum)) an estimate of the currentminimum range at any time t_(i) make be found using the followingformula:R_((CURRENTMINIMUM))=R_(CPA(MINIMUM))/cos(θ₀−θ_(i))  (14)

In an exemplary embodiment, the above may be used to base targetopen-close on measurements calculated at the time of the decision.

Referring now to FIG. 3, a Cartesian graph of target maneuvers andnoise, if more than three points are used, a series of subsequentmeasurements may be used to determine maneuvering of target 2. By way ofexample and not limitation, a set of five or more usable bearing pointsmay be obtained as a set of calculated points C₁, C₂, and C₃ inaccordance with the teachings of the present inventions during times{t₁,t₂,t₃}, {t₂,t₃,t₄}, and {t₃,t₄,t₅} (these time points are not shownin FIG. 3). Points C₁, C₂, and C₃ may be extrapolated to indicate thattarget 2 (shown as the dark circles in FIG. 3) is maneuvering in anon-linear fashion.

Additionally, the estimates may be used to determine noise or a range ofnoise in the readings. By way of example and not limitation, a set offive or more usable bearing points may be interpreted as a set ofcalculated points P₁, P₂, and P₃ obtained in accordance with theteachings of the present inventions during times {t₆,t₇,t₈}, {t₇,t₈,t₉},and {t₈,t₉,t₁₀} (these time points are not shown in FIG. 3). However, P₂can be seen to have deviated from a predicted point P₂′, indicating thatnoise is present in the system. In a currently envisioned embodiment,trends over time may therefore use these deviations to estimate theamount and effects of noise present in the system. If an assumption ismade that any set of four points represents a stable, noise-freesolution, analysis of deviation from a predicted point may be made withfour points. In such an analysis, a fifth point may then be obtained andused to determine if the deviation is random or the result of adeterministic event, e.g. a maneuvering of target 2. Thus, a minimum setof points required to detect the possible presence of noise is four, andthe minimum set of points required to detect the possible presence ofmaneuvering of target 2 is five.

Referring back to FIG. 2, in a reference frame 200 relative to aposition of ownship 1, three bearing/time measurements are taken, anangle to bearing at CPA relative to a heading of ownship 1 iscalculated, and the time of CPA is calculated. Based on the teachings ofthese inventions that target 2 and ownship 1 remain on a constant courseand speed over a period of time required to collect bearingmeasurements, a fourth data point may be obtained. When taken with anyof the other two of the three bearing data points, the fourth data pointshould yield the same solution, i.e., the angle to bearing at CPArelative to the heading of ownship 1, and the time of CPA will beconstant for all combinations of the three of four bearing data points.A deviation in the bearing at CPA relative to the heading of ownship 1and the time of CPA represents noise in the system which can be detectedby this method of calculating the angle to bearing at CPA for eachpotential solution.

Prior art methods look at each bearing measurement as a unique point in“the” solution set and do not consider triplet-wise combinations ofpoints as potential solutions to the angle at CPA, each one as valid asthe other, if the bearing measurements are independent. Therefore, withthe present inventions, with four data points, four potential solutionsmay be investigated; with five independent points, ten potentialsolutions may be investigated; and with six independent points, twentypotential solutions may be investigated. This is quickly recognized asthe number of possible combinations of n items taken three at a time. Astatistical analysis of the potential solutions may then yield trendsand/or the mean and standard deviation of bearings at CPA. The mean ofthe bearing at CPA and the mean time of CPA are more accurate solutionsof the bearing at CPA and time of CPA than any one potential solutionbased on a triplet of bearing measurements.

Thus, the present inventions may allow creating twenty solutions withonly six data points rather than waiting for twenty data points.Likewise, four points may be sufficient to determine that there is noisein system and calculating four bearing angle solutions at CPA provides afirst order estimate of the magnitude of the noise and a first orderestimate of the mean bearing at CPA and mean time of CPA.

It is also noted that in the preferred embodiment, bearing rate curveinflection points are always plus or minus around 30° of the BRG at CPA.

It will be understood that various changes in the details, materials,and arrangements of the parts which have been described and illustratedabove in order to explain the nature of this inventions may be made bythose skilled in the art without departing from the principle and scopeof the inventions as recited in the following claims.

1. A method of estimating a minimum range from an ownship to a target ata closest point of approach (CPA) between the target and the ownship,comprising: a.a bearing detector obtaining at least three bearing datapoints of the target with respect to anthe ownship, wherein each of saidbearing data points includes a bearing angle and a corresponding time ofacquisition; b. using the three bearing data points to determine a speedcontribution V_(os) of a first point of interest to a distance from arelative velocity vector over a time frame comprising an initial timet_(o) to a predetermined time t_(i); c.a computer system determining anangle θ_(β)as defined as, where θ_(β) is the bearing relative to theownship's heading at the point in time (t_(β)) of the closest point ofapproach to a second point of interest; and d.the computer systemcalculating a minimum range Min R_(CPA) using the formula:Min R_(CPA)=V_(OS)(t_(β)−t_(i))cos(θ_(β)−θ_(i))_(θ) _(i) _(|=0); usingthe calculated minimum range for at least one of: targeting a weaponwith respect to the target, navigating the ownship; e. wherein t_(β) isthe time at which θ_(β) was mreaured and θ_(i) is a bearing angle to thetarget relative to the ownship corresponding to a first of said at leastthree bearing data points obtained at time t_(i), and V_(os) is thespeed of the ownship during said obtaining said at least three bearingdata points.
 2. The method of claim 1, further comprising generating arepresentation of the probability of the location of the target usingthe calculated minimum range.
 3. The method of claim 1, wherein thecalculated minimum range is further used for at least one of targeting aweapon with respect to the second point of interest, navigation of theownship, estimating a passing range between the ownship and the secondpoint of interest target, and avoidance of the second point of interest.4. The method of claim 1, wherein the at least three bearing data pointsare obtained passively.
 5. The method of claim 1, further comprising: f.obtaining a fourth bearing data point of the second point with respectto an ownship; g. calculating a further set of minimum ranges using theformula of step (d) for Min R_(CPA); and h. repeating steps (e) and (f)obtaining bearing data points and performing corresponding calculationsof Min R_(CPA) to determine a maneuvering of the second point ofinterest target over time.
 6. The method of claim 1, further comprising:f. obtaining an additional plurality of bearing data points of thesecond point target with respect to an the ownship; g. calculating afurther set of minimum ranges using the formula of step (d) for MinR_(CPA); and h. determining a deviation of a calculated minimum rangefrom others of the calculated minimum ranges.
 7. A method for estimatinga minimum range Min R_(CPA) to a contact from an ownship, independent ofactual contact range, comprising: a. a bearing detector passivelyobtaining at least three bearing data points of the contact relative toan the ownship; b. a computer system determining an angle θ_(β) definingthe bearing to the contact relative to a heading of the ownship at thepoint in time of closest approach to a second point of interest thecontact; c. the computer system calculating a the minimum range at CPA aclosest point of approach (CPA) between the ownship and the targetcontact using the formulaMin R_(CPA)=V_(os)(t_(β)−t_(i))cos(θ_(β)−θ_(i))_(θ) _(i) _(|=0); and d.generating a representation of the probability of the location of thetarget contact located at the minimum range; d. using the calculatedminimum range to alter a heading of the ownship; e. wherein t_(β) is thetime at which corresponding to θ_(β)was measured, θ_(i) is a bearingangle to the contact relative to the ownship at time t_(i); and V_(os)is a speed contribution of a first point of interest to a distance froma relative velocity vector over a time frame comprising an initial timet₀ to a predetermined time t_(i) the ownship during said passivelyobtaining said at least three bearing data points.
 8. The method ofclaim 7, further comprising: f. obtaining a fourth data point; g. usingthe fourth data point to calculate an angle to bearing at CPA relativeto the heading of the ownship; h. calculating a time of CPA for allcombinations of the three of four bearing data points; and i.determining noise in the system by comparing a deviation in at least oneof the bearing at CPA, relative to the heading of the ownship and thetime of CPA for each potential solution, to a predetermined value. 9.The method of claim 8, wherein the step of determining noise in thesystem further comprises determining the mean and standard deviations inthe bearing calculations at CPA.
 10. The method of claim 7 furthercomprising: f. obtaining an estimate of a current minimum range at atime t_(i), the estimate comprising: i. calculating a current minimumrange R_((current minimum)) by dividing Min R_(CPA) by the cosine of(θ_(β)−θ_(i)) where θ₀ is a bearing relative to the ownship when θ=0,and θ_(i) is a bearing relative to the ownship at time t_(i); and ii.generating a representation of the probability of the location of thecontact.
 11. The method of claim 7, further comprising: f. obtainingsaid additional bearing data points of the second point of interestcontact with respect to said ownship; g. using the additional bearingdata points to refine the system noise estimate by calculating the meanand standard deviation of the bearings at CPA; h. using the additionalbearing data points to refine the mean bearing at CPA with respect toownship's heading; i. determining a trend of change in the mean value ofbearing at CPA with respect to ownship's heading; j. using the trend ofchange in the mean value of bearing at CPA with respect to ownship'sheading to determine change in a relative velocity vector between saidownship and said target contact.
 12. A system for calculating anestimated minimum range estimate R_(CPA) from a source to a target,comprising: a. a bearing detector capable of passively obtaining abearing to the target from the source; b. a computer having a processorand memory; and c. range calculation software executing in the computer;d. wherein i. the memory stores at least three bearing data pointsobtained from the bearing detector; ii. the range calculation softwareuses the stored three bearing data points to determine a speedcontribution V_(os) of the target to a distance from a relative velocityvector over source during a time from t₀ to t₀′ when said at least threebearing data points are obtained; iii. the range calculation softwaredetermines an angle θ_(β) defined by the bearing to the target relativeto a heading of the source at the point in time of closest approach tobetween the source and the target; iv. the range calculation softwarecalculates a minimum range from the source to the targetand as MinR_(CPA)=V_(OS)(t_(β)-t_(i))cos(θ_(β)θ_(i))_(θi|=0); and, wherein saidminimum range is based in part on V_(os), θ_(β), and the point in timeof closest approach; and v. the range calculation software generates arepresentation of the probability of the location of a target. whereinthe system is configured to use the calculated minimum range to alter aheading of the source; wherein the source and the target are physicalobjects.
 13. The system of claim 12 further comprising an output devicecapable of reproducing a representation of at least one of thecalculated minimum range output and the probability of the location ofthe target.
 14. A method, comprising: a. a bearing detector obtaining atleast three bearing data points of a target with respect to a vehicle;b. a computer system determining an angle θ_(β), wherein θ_(β) isdefined as the bearing of the target relative to the vehicle's headingat the time of closest approach to the target; c. the computer systemestimating a minimum range from the vehicle to the target using saidobtained three bearing data points, said bearing angle θ_(β) and a speedof the vehicle during said obtaining; and d. using said estimatedminimum range to alter a heading of the vehicle.
 15. The method of claim1, further comprising using said calculated minimum range at the closestpoint of approach to estimate a minimum range at time t_(i).
 16. Themethod of claim 15, wherein said minimum range at said time t_(i) isequal to Min R_(CPA) divided by cos(θ₀−θ_(i)), wherein θ₀ is a bearingangle at time t₀ and θ_(i) is a bearing angle at said time t_(i). 17.The method of claim 1, wherein θ_(β) is calculated according to thefollowing formula:${\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{j} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{k} \right)}{\Delta t}_{i,j}}}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}{\Delta t}_{i,j}}} \right\rbrack}};$wherein θ_(j) and θ_(k) are bearing angles respectively corresponding tosecond and third ones of said at least three bearing data points,wherein θ_(j) and θ_(k) are obtained at times t_(j) and t_(k)respectively, and wherein Δt_(j,k), Δt_(k,i), Δt_(i,j) are thedifferences between times t_(j) and t_(k); t_(k) and t_(i); and t_(i)and t_(j), respectively.
 18. The method of claim 7, further comprisingusing said calculated minimum range at the closest point of approach toestimate a minimum range at time t_(i).
 19. The method of claim 18,wherein said minimum range at said time t_(i) is equal to Min R_(CPA)divided by cos(θ₀−θ_(i)), wherein θ₀ is a bearing angle at time t₀ andθ_(i) is a bearing angle at said time t_(i).
 20. The method of claim 7,wherein θ_(β) is calculated according to the following formula:${\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{j} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{k} \right)}{\Delta t}_{i,j}}}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}{\Delta t}_{i,j}}} \right\rbrack}};$wherein θ_(j) and θ_(k) are bearing angles respectively corresponding tosecond and third ones of said at least three bearing data points,wherein θ_(j) and θ_(k) are obtained at times t_(j) and t_(k)respectively, and wherein Δt_(j,k), Δt_(k,i), Δt_(i,j) are thedifferences between times t_(j) and t_(k); t_(k) and t_(i); and t_(i)and t_(j), respectively.
 21. A method for tracking a second point ofinterest relative to a first point of interest, said method comprising:a computer system receiving information indicative of at least threebearing data points of said second point of interest relative to saidfirst point of interest, wherein each of the at least three bearing datapoints includes a bearing angle and a corresponding acquisition time,wherein each acquisition time is different; the computer systemestimating a minimum range of said second point of interest relative tosaid first point of interest, wherein said estimating uses one or moreequations, wherein said one or more equations have a closed-formsolution, and wherein at least one of said one or more equations isbased in part upon three of said at least three bearing data points; andaltering a heading of the first point of interest based at least in parton the estimated minimum range; wherein the first and second points ofinterest are physical objects.
 22. The method of claim 21, wherein atleast one of said one or more equations is also based in part on a speedof said first point of interest.
 23. The method of claim 22, whereinsaid estimated minimum range corresponds to a closest point of approach(CPA) between the first and second points of interest.
 24. The method ofclaim 23, further comprising using said estimated minimum rangecorresponding to said CPA to estimate a minimum range at a time t_(i).25. The method of claim 24, wherein said minimum range at said timet_(i) is equal to said minimum range corresponding to said CPA dividedby cos(θ₀−θ_(i)), wherein θ₀ is a bearing angle at a time t₀ and θ_(i)is a bearing angle at said time t_(i).
 26. The method of claim 23,wherein said estimating said minimum range includes estimating a bearingangle θ_(β) at the CPA.
 27. The method of claim 26, wherein saidestimating θ_(β) is based in part upon said at least three bearing datapoints.
 28. The method of claim 26, wherein θ_(β) is calculated usingthe following equation:${\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{j} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{k} \right)}{\Delta t}_{i,j}}}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}{\Delta t}_{i,j}}} \right\rbrack}};$wherein θ_(j) and θ_(k) are bearing angles respectively corresponding tosecond and third ones of said at least three bearing data points,wherein θ_(j) and θ_(k) are obtained at times t_(j) and t_(k)respectively, and wherein Δt_(j,k), Δt_(k,i), Δt_(i,j) are thedifferences between times t_(j) and t_(k); t_(k) and t_(i); and t_(i)and t_(j), respectively.
 29. The method of claim 23, wherein theestimation of said minimum range is based upon a time t_(β)corresponding to the CPA.
 30. The method of claim 21, wherein saidminimum range (Min R_(CPA)) corresponds to a closest point of approach(CPA) between the first and second points of interest, and wherein MinR_(CPA) is calculated according to the formula MinR_(CPA)=V_(os)(t_(β)−t_(i))cos(θ_(β)−θ_(i))_(θi|=0), and wherein V_(os)is a speed of said first point of interest, θ_(i) is a bearing anglebetween the first point of interest and the second point of interest attime t_(i), and θ_(β) is a bearing angle between the first point ofinterest and the second point of interest at time t_(β), wherein t_(β)is an estimated time corresponding to the CPA.
 31. The method of claim21, wherein said at least three bearing data points include four or morebearing data points, the method further comprising estimating a minimumrange corresponding to each three data point-combination of the four ormore bearing data points.
 32. The method of claim 31, further comprisingperforming a statistical analysis on each of said estimated minimumranges.
 33. The method of claim 32, wherein said statistical analysisincludes calculating a mean minimum range.
 34. The method of claim 32,wherein said statistical analysis includes calculating a standarddeviation of said estimated minimum range.
 35. The method of claim 21,wherein said receiving includes receiving four or more bearing datapoints, the method further comprising using the received four or moredata points to detect the presence of noise.
 36. The method of claim 21,wherein said receiving includes receiving five or more bearing datapoints, the method further comprising using the received five or moredata points to detect maneuvering of said second point of interest. 37.The method of claim 21, wherein said first point of interest is a watervessel.
 38. The method of claim 21, wherein said second point ofinterest is a water vessel.
 39. The method of claim 21, wherein saidfirst point of interest is in motion, and said second point of interestis stationary.
 40. The method of claim 21, wherein the one or moreequations include the following mathematical operations: addition,subtraction, multiplication, division, cosine, tangent, inverse tangent.41. The method of claim 21, further comprising using said estimatedminimum range to launch a weapon at said second point of interest.
 42. Amethod for tracking a second point of interest relative to a first pointof interest, said method comprising: a computer system receivinginformation indicative of at least three bearing data points, whereineach of said at least three bearing data points includes a bearing angleand a corresponding acquisition time, wherein each bearing angle ismeasured between a heading of said first point of interest and thesecond point of interest at said corresponding acquisition time, whereineach said corresponding acquisition time is different; the computersystem estimating a minimum range of said second point of interestrelative to said first point of interest, wherein said estimating isperformed in a single iteration through a set of one or more equations,wherein said set of equations are based in part upon three of said atleast three bearing data points; and altering a heading of the firstpoint of interest based at least in part on the estimated minimum range;wherein the first and second points of interest are physical objects.43. The method of claim 42, wherein said set of equations are based inpart upon a speed of the first point of interest.
 44. The method ofclaim 42, wherein said at least three bearing data points is a number(N) of bearing data points greater than or equal to four, said methodfurther comprising performing a number (C) of minimum range calculationsfor each three data point-combination of said N bearing data points,where C=N!/((N−3)!*3!), wherein each of said C minimum rangecalculations is performed in a single iteration through said set ofequations.
 45. The method of claim 42, further comprising computing amean minimum range from said C minimum range calculations.
 46. Themethod of claim 42, further comprising computing a standard deviation ofsaid C minimum range calculations.
 47. The method of claim 42, whereineither or both of said first and second points of interest are watervessels.
 48. The method of claim 42, wherein said set of equations isbased in part upon an angle between a heading of said first point ofinterest and said second point of interest at a closest point ofapproach between said first and second points of interest.
 49. Themethod of claim 42, further comprising using said estimated minimumrange to alter a heading of said first point of interest.
 50. The methodof claim 42, further comprising using said estimated minimum range tolaunch a weapon at said second point of interest.
 51. The method ofclaim 42, wherein said minimum range corresponds to a closest point ofapproach between said first and second point of interest.
 52. The methodof claim 42, further comprising using said minimum range at said closestpoint of approach to calculate a minimum range at a different time. 53.A system, comprising: a processor; and a memory coupled to theprocessor, wherein the memory is configured to store programinstructions executable by the processor to: receive at least threebearing data points of a second point of interest relative to a firstpoint of interest, wherein each of the at least three bearing datapoints includes a bearing angle and a corresponding acquisition time,wherein each bearing angle is an angle between a heading of said firstpoint of interest and a second point of interest at said correspondingacquisition time, wherein each acquisition time is different, andwherein said first and second points of interest are physical objects;and estimate a minimum range of said second point of interest relativeto said first point of interest, wherein said estimation uses one ormore equations, wherein said one or more equations have a closed-formsolution, and wherein at least one of said one or more equations isbased in part upon three of said at least three bearing data points;wherein said system is further configured to use said estimated minimumrange to alter a heading of said first point of interest.
 54. The systemof claim 53, further comprising one or more bearing detectors configuredto obtain bearing data points.
 55. The system of claim 54, wherein saidbearing detectors are configured to obtain said bearing data pointspassively.
 56. The system of claim 53, wherein the one or more equationsinclude the following mathematical operations: addition, subtraction,multiplication, division, cosine, tangent, inverse tangent.
 57. Thesystem of claim 53, wherein said system is further configured to usesaid estimated minimum range to target said second point of interestusing a weapons system configured to target said second point ofinterest.
 58. The system of claim 53, wherein said estimated minimumrange corresponds to a closest point of approach (CPA) between saidfirst and second points of interest, and wherein said system is furtherconfigured to use said estimated minimum range in order to estimate aminimum range at a time other than a time corresponding to said CPA. 59.A system, comprising: a processor; and a memory coupled to theprocessor, wherein the memory is configured to store programinstructions executable by the processor to: receive at least threebearing data points of a second point of interest relative to a firstpoint of interest, wherein said data points are acquired at differenttimes, and wherein said first and second points of interest are physicalobjects; and estimate a minimum range of said second point of interestrelative to said first point of interest, wherein said estimating isperformed in a single iteration through a set of one or more equations,wherein said set of equations are based in part upon three of said atleast three bearing data points; wherein said system is furtherconfigured to use said estimated minimum range to target said secondpoint of interest with a weapons system.
 60. The system of claim 59,wherein each of the at least three bearing data points includes abearing angle and a corresponding acquisition time.
 61. The system ofclaim 59, further comprising one or more bearing detectors configured toobtain bearing data points.
 62. The system of claim 59, wherein thenumber of said at least three bearing data points is a number (N)greater than or equal to four, said method further comprising performinga number (C) of minimum range calculations for each three datapoint-combination of said N bearing data points, where C=N!/((N−3)!*3!),wherein each of said C minimum range calculations is performed in asingle iteration through said set of equations.
 63. The system of claim59, wherein said system is further configured to use said estimatedminimum range to alter a heading of said first point of interest.
 64. Anon-transitory computer readable medium comprising program instructions,wherein the instructions are computer-executable to: receive at leastthree bearing data points of a second point of interest relative to afirst point of interest, wherein each of the at least three bearing datapoints includes a bearing angle and a corresponding acquisition time,wherein each acquisition time is different; estimate a minimum range ofsaid second point of interest relative to said first point of interest,wherein said estimation uses one or more equations, wherein said one ormore equations have a closed-form solution, and wherein said one or moreequations are based in part upon three of said at least three bearingdata points; and use said estimated minimum range to alter a heading ofsaid first point of interest; wherein said first and second points ofinterest are physical objects.
 65. The non-transitory computer readablemedium of claim 64, wherein the one or more equations include thefollowing mathematical operations: addition, subtraction,multiplication, division, cosine, tangent, inverse tangent.
 66. Anon-transitory computer readable medium comprising program instructions,wherein the instructions are computer executable to: receive at leastthree bearing data points of a second point of interest relative to afirst point of interest, wherein each of said data points corresponds todifferent points in time, and wherein said first and second points ofinterest are physical objects; calculate an estimation of a minimumrange of said second point of interest relative to said first point ofinterest, wherein said estimation is performed in a single iterationthrough one or more equations, wherein said one or more equations dependin part upon three of said at least three bearing data points; and usesaid estimated minimum range to target said second point of interestwith a weapons system.
 67. A method, comprising: a computer systemreceiving information indicative of at least three bearing data points,wherein each of the at least three bearing data points includes abearing angle and a corresponding acquisition time, wherein each bearingangle is measured between a heading of a first point of interest and asecond point of interest, and wherein each acquisition time isdifferent; the computer system estimating a minimum range of said secondpoint of interest relative to said first point of interest, wherein saidestimating is based on one or more equations having a closed-formsolution, and wherein said one or more equations are based in part uponthree of said at least three bearing data points; and using saidestimated minimum range to change a heading of said first point ofinterest; wherein said first point of interest and said second point ofinterest are physical objects, and wherein said first point of interestis a vehicle.
 68. The method of claim 67, wherein said first point ofinterest is an automobile.
 69. The method of claim 67, wherein saidfirst point of interest is a water vessel.
 70. The method of claim 67,wherein said first point of interest is an aircraft.
 71. The method ofclaim 67, wherein said estimation of said minimum range is based in partupon a bearing angle θ_(β) that corresponds to a closest point ofapproach (CPA) between said first and second points of interest.
 72. Themethod of claim 67, wherein said bearing angle θ_(β) at the CPA iscalculated according to the following formula:${\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{j} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{k} \right)}{\Delta t}_{i,j}}}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}{\Delta t}_{i,j}}} \right\rbrack}};$wherein θ_(j) and θ_(k) are bearing angles respectively corresponding tosecond and third ones of said at least three bearing data points,wherein θ_(j) and θ_(k) are obtained at times t_(j) and t_(k)respectively, and wherein Δt_(j,k), Δt_(k,i), Δt_(i,j) are thedifferences between times t_(j) and t_(k); t_(k) and t_(i); and t_(i)and t_(j), respectively.
 73. The method of claim 67, wherein saidestimated minimum range corresponds to a closest point of approach (CPA)between said first and second points of interest, and wherein saidmethod further comprises using said estimated minimum range at said CPAto estimate a minimum range at a time t_(i).
 74. The method of claim 73,wherein said minimum range at said time t_(i) is equal to said minimumrange at said CPA divided by cos(θ₀−θ_(i)), wherein θ₀ is a bearingangle at time t₀ and θ_(i) is a bearing angle at said time t_(i).
 75. Amethod, comprising: a computer system receiving information indicativeof at least three bearing data points, wherein each of the at leastthree bearing data points includes a bearing angle and a correspondingacquisition time, wherein each bearing angle is measured between aheading of a first point of interest and a second point of interest, andwherein each acquisition time is different, and wherein said first andsecond points of interest are physical objects; the computer systemestimating a minimum range of said second point of interest relative tosaid first point of interest, wherein said estimating is based on one ormore equations having a closed-form solution, and wherein said one ormore equations are based in part upon three of said at least threebearing data points; and targeting said second point of interest using aweapons system, wherein said targeting is based in part upon saidestimated minimum range.
 76. The method of claim 75, wherein said firstpoint of interest is a water vessel.
 77. The method of claim 75, whereinsaid estimation of said minimum range is based in part upon a bearingangle θ_(β) that corresponds to a closest point of approach (CPA)between said first and second points of interest.
 78. The method ofclaim 77, wherein said bearing angle θ_(β) at the CPA is calculatedaccording to the following formula:${\left( \theta_{\beta} \right) = {\tan^{- 1}\left\lbrack \frac{{{\tan\left( \theta_{i} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{j} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{k} \right)}{\Delta t}_{i,j}}}{{{\tan\left( \theta_{j} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{j,k}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{k} \right)}{\Delta t}_{k,i}} + {{\tan\left( \theta_{i} \right)}{\tan\left( \theta_{j} \right)}{\Delta t}_{i,j}}} \right\rbrack}};$wherein θ_(j) and θ_(k) are bearing angles respectively corresponding tosecond and third ones of said at least three bearing data points,wherein θ_(j) and θ_(k) are obtained at times t_(j) and t_(k)respectively, and wherein Δt_(j,k), Δt_(k,i), Δt_(i,j) are thedifferences between times t_(j) and t_(k); t_(k) and t_(i); and t_(i)and t_(j), respectively.
 79. The method of claim 75, wherein saidestimated minimum range corresponds to a closest point of approach (CPA)between said first and second points of interest, and wherein saidmethod further comprises using said estimated minimum range at said CPAto estimate a minimum range at a time t_(i).
 80. The method of claim 75,wherein said minimum range at said time t_(i) is equal to said minimumrange at said CPA divided by cos(θ₀−θ_(i)), wherein θ₀ is a bearingangle at time t₀ and θ_(i) is a bearing angle at said time t_(i). 81.The method of claim 14, wherein the vehicle is an aircraft, a watervessel, or an automobile.
 82. The method of claim 67, wherein the secondpoint of interest is another vehicle.
 83. The method of claim 67,wherein the second point of interest is a stationary object.